A pilot flies her route in two straight-line segments. The displacement vector A for the first segment has a magnitude of 254 km and a direction 30.0o north of east. The displacement vector for the second segment has a magnitude of 178 km and a direction due west. The resultant displacement vector is R = A + B and makes an angle è with the direction due east. Using the component method, find (a) the magnitude of R and (b) the directional angle è.
I know that for the magnitude, since both vectors have different directions I cannot just add them together. I am not sure where to go from here.
Divide both vectors into East (x) and North (y) components
First vector:
east1 = 254 cos 30 = 220
north1 =254 sin 30 = 127
Second vector
east 2 = - 178
north2 = 0
now add east components
220 - 178 = 42 east
now the north components
127 + 0 = 127
so
magnitude = sqrt (42^2+127^2) = 134
tan e = 127/42 = 3.02
so e = 71.7 degrees north of east
To find the magnitude of the resultant vector R, we can break down both vectors A and B into their horizontal and vertical components.
For vector A, the angle of 30.0 degrees north of east means that it can be broken down into two components: one along the x-axis (east) and one along the y-axis (north).
The x-component of vector A (Ax) can be found using trigonometry:
Ax = magnitude of A * cos(angle) = 254 km * cos(30.0 degrees) = 220.174 km (rounded to three decimal places)
The y-component of vector A (Ay) can also be found using trigonometry:
Ay = magnitude of A * sin(angle) = 254 km * sin(30.0 degrees) = 127 km (rounded to the nearest kilometer)
For vector B, since it is directed due west, it only has a horizontal component (Bx) and no vertical component (By).
Therefore, Bx = magnitude of B = 178 km
Now we can find the horizontal and vertical components of the resultant vector R by adding the corresponding components of A and B:
Rx = Ax + Bx = 220.174 km + 178 km = 398.174 km (rounded to three decimal places)
Ry = Ay + By = 127 km + 0 km = 127 km
The magnitude of the resultant vector R can be found using the Pythagorean theorem:
|R| = sqrt(Rx^2 + Ry^2) = sqrt((398.174 km)^2 + (127 km)^2) = 423.119 km (rounded to three decimal places)
Therefore, the magnitude of R is approximately 423.119 km.
To find the directional angle θ, we can use inverse tangent (tan^(-1)) with the horizontal and vertical components of R:
θ = tan^(-1)(Ry/Rx) = tan^(-1)(127 km/398.174 km) ≈ 18.168 degrees (rounded to three decimal places)
Thus, the directional angle θ is approximately 18.168 degrees.
To find the magnitude of the resultant displacement vector R, you can use the component method. Here are the steps to follow:
Step 1: Convert the given directions to x and y components.
- For the first segment, the direction is 30.0° north of east. We can split this into x and y components as follows:
- The x component (horizontal) is A1 = A * cos(30.0°)
- The y component (vertical) is A2 = A * sin(30.0°)
- For the second segment, the direction is due west, which means it is pointing in the negative x direction. The x component is B1 = -B, and the y component remains 0 since there is no vertical displacement.
Step 2: Add the x and y components separately to find the resultant x and y components.
- R1 = A1 + B1
- R2 = A2 + B2
Step 3: Calculate the magnitude of the resultant vector R using the Pythagorean theorem.
- R = sqrt(R1^2 + R2^2)
Step 4: Calculate the directional angle θ of R with respect to the positive x-axis using the inverse tangent function.
- θ = atan(R2 / R1)
Now let's proceed with the calculations:
Given:
A = 254 km
B = 178 km
Angle A makes with the positive x-axis = 30.0 degrees
Step 1:
A1 = A * cos(30.0°) = 254 km * cos(30.0°) = 219.1 km
A2 = A * sin(30.0°) = 254 km * sin(30.0°) = 127 km
B1 = -B = -178 km
B2 = 0 km
Step 2:
R1 = A1 + B1 = 219.1 km + (-178 km) = 41.1 km
R2 = A2 + B2 = 127 km + 0 km = 127 km
Step 3:
R = sqrt(R1^2 + R2^2) = sqrt((41.1 km)^2 + (127 km)^2) = sqrt(1690.21 km^2 + 16129 km^2) = sqrt(17819.21 km^2) ≈ 133.5 km
Step 4:
θ = atan(R2 / R1) = atan(127 km / 41.1 km) = atan(3.092) ≈ 72.4°
Therefore, the magnitude of R is approximately 133.5 km, and the directional angle è is approximately 72.4° with respect to the positive x-axis.